Due to the mutual dependency of the two equations (REF )-(REF ), we have to find a fixed point between the two equations. The existence of it has been largely discussed in various situations [1]}, [2]} but never in the hybrid control framework. Some partial results are described in [3]} for an optimal stopping problem. To derive the fixed point numerically, the literature proposes two possible options: the use of a global descent method between the two-equation [4]}, or a fixed point iteration between the two equations [5]}, [6]}. We opted for the second one in this test, iterating between the two equations till a stopping criterium is verified. We use as stop criterium the distance in discrete infinite norm between two iterations of the density
\( {\mathcal {E}}(k)=\Vert \mu ^z-\mu ^{z-1}\Vert =\max _{k,i}|\mu ^z(x_i, t_k)-\mu ^{z-1}(x_i, t_k)|.\) 
